A006506 -id:A006506 - cp's OEIS frontend

A068471 Number of n X n binary matrices with each 1 having at most 2 adjacent 1's.

1, 2, 16, 378, 30824, 8402216, 7664347268, 23371379782671, 238225926162821893, 8118262028301675826132, 924887563235974860108746534, 352261845112790535941917078458268

Offset: 0

R. H. Hardin, Mar 10 2002

nonn

No adjacent 1's A006506, one adjacent 1 A067968.

a(0)=1 prepended by Alois P. Heinz, Sep 26 2024

A201222 Number of ways to place k non-attacking knights on a 2 X n horizontal cylinder, summed over all k>=0.

Original entry on oeis.org

3, 9, 18, 81, 123, 324, 843, 2401, 5778, 15129, 39603, 104976, 271443, 710649, 1860498, 4879681, 12752043, 33385284, 87403803, 228886641, 599074578, 1568397609, 4106118243, 10750371856, 28143753123, 73681302249, 192900153618, 505022001201, 1322157322203

Offset: 1

Vaclav Kotesovec, Nov 28 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Non-attacking chess pieces

Cf. A189145, A001333, A051927, A006506, A067966.

Table[If[Mod[n,4]==0,LucasL[n/2]^4,LucasL[2n]+1+(-1)^n],{n,1,50}]

A283161 Natural numbers whose digits can be formed by typing non-adjacent keys on a 123-456-789 keypad without repeating a digit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 15, 16, 17, 18, 19, 24, 26, 27, 28, 29, 31, 34, 35, 37, 38, 39, 42, 43, 46, 48, 49, 51, 53, 57, 59, 61, 62, 64, 67, 68, 71, 72, 73, 75, 76, 79, 81, 82, 83, 84, 86, 91, 92, 93, 94, 95, 97, 135, 137, 138, 139, 153, 157, 159, 167, 168, 173, 175, 176, 179, 183, 186, 193, 195

Offset: 1

FUNG Cheok Yin, Mar 02 2017

Or say numbers which are "very difficult" to be typed on a keypad without the zero. (See description of A215009.)

			The keypad is:
+-----+
|1|2|3|
+-+-+-+
|4|5|6|
+-+-+-+
|7|8|9|
+-+-+-+
It is visibly obvious that 168 can be formed on the keypad, and each pairwise digits of 168 are not adjacent.

FUNG Cheok Yin, Table of n, a(n) for n = 1..453
FUNG Cheok Yin, C++ program

Cf. A280593, A215009, A006506(3)

no = IntegerDigits @ {12,14,23,25,34,36,45,47,56,58,69,78,89}; Sort[ FromDigits /@ Flatten[ Permutations /@ Select[ Subsets[ Range@ 9, {1, 9}], Intersection[ Subsets[#, {2}], no] == {} &], 1]] (* Giovanni Resta, Apr 06 2017 *)

A350336 Number of n X n ternary matrices with no two adjacent 0's.

Original entry on oeis.org

1, 3, 56, 7504, 6832640, 42780151808, 1836366011301888, 540795841280638713856, 1092417949346109029345132544, 15137179876232766647722798101823488, 1438787206346713875314130065804001328234496, 938091111277955250977701268973340995182098116509696

Offset: 0

Robert P. P. McKone, Jan 03 2022

A two-dimensional generalization of A028859.

2^(n^2) < a(n) < 3^(n^2).

			a(1) is trivial because all 3 1 X 1 matrices have no 2 adjacent 0's, whereas for a(2) the 56 matrices are:
  {
    {{0, 1}, {1, 0}}, {{0, 1}, {1, 1}},
    {{0, 1}, {1, 2}}, {{0, 1}, {2, 0}},
    {{0, 1}, {2, 1}}, {{0, 1}, {2, 2}},
    {{0, 2}, {1, 0}}, {{0, 2}, {1, 1}},
    {{0, 2}, {1, 2}}, {{0, 2}, {2, 0}},
    {{0, 2}, {2, 1}}, {{0, 2}, {2, 2}},
    {{1, 0}, {0, 1}}, {{1, 0}, {0, 2}},
    {{1, 0}, {1, 1}}, {{1, 0}, {1, 2}},
    {{1, 0}, {2, 1}}, {{1, 0}, {2, 2}},
    {{1, 1}, {0, 1}}, {{1, 1}, {0, 2}},
    {{1, 1}, {1, 0}}, {{1, 1}, {1, 1}},
    {{1, 1}, {1, 2}}, {{1, 1}, {2, 0}},
    {{1, 1}, {2, 1}}, {{1, 1}, {2, 2}},
    {{1, 2}, {0, 1}}, {{1, 2}, {0, 2}},
    {{1, 2}, {1, 0}}, {{1, 2}, {1, 1}},
    {{1, 2}, {1, 2}}, {{1, 2}, {2, 0}},
    {{1, 2}, {2, 1}}, {{1, 2}, {2, 2}},
    {{2, 0}, {0, 1}}, {{2, 0}, {0, 2}},
    {{2, 0}, {1, 1}}, {{2, 0}, {1, 2}},
    {{2, 0}, {2, 1}}, {{2, 0}, {2, 2}},
    {{2, 1}, {0, 1}}, {{2, 1}, {0, 2}},
    {{2, 1}, {1, 0}}, {{2, 1}, {1, 1}},
    {{2, 1}, {1, 2}}, {{2, 1}, {2, 0}},
    {{2, 1}, {2, 1}}, {{2, 1}, {2, 2}},
    {{2, 2}, {0, 1}}, {{2, 2}, {0, 2}},
    {{2, 2}, {1, 0}}, {{2, 2}, {1, 1}},
    {{2, 2}, {1, 2}}, {{2, 2}, {2, 0}},
    {{2, 2}, {2, 1}}, {{2, 2}, {2, 2}}
  }

Andrew Howroyd, Table of n, a(n) for n = 0..30

Cf. A006506 for binary version.

Cf. A028859 for one-dimensional version.

t[m_] := t[m] = Map[ArrayReshape[#, {m, m}] &, Tuples[{0, 1, 2}, m^2]];a[m_] := a[m] = Count[Table[AnyTrue[Flatten[{Table[Equal[0, t[m][[n, a, b]], t[m][[n, a, b + 1]]], {a, 1, m}, {b, 1, m - 1}], Table[Equal[0, t[m][[n, a, b]], t[m][[n, a + 1, b]]], {a, 1, m - 1}, {b, 1, m}]}], TrueQ], {n, 1, 3^(m^2)}], False]; Table[a[n], {n, 1, 3}]

Terms a(5)-a(11) from Andrew Howroyd, Jan 04 2022

A381857 Number of n X n binary matrices with at least 2 adjacent 1's.

Original entry on oeis.org

0, 0, 9, 449, 64302, 33498985, 68713877875, 562948673292362, 18446743413061588661, 2417851638458709952150645, 1267650600226199352445557225326, 2658455991569819662405962686908743173, 22300745198530622979053904922855772969397419

Offset: 0

Benjamin Ghitterman, Mar 08 2025

nonn

Use black and white square tiles to fill an n X n grid. In some combinations, a black tile will share a side with another black tile. This sequence counts the number of arrangements containing at least one black tile sharing a side with another black tile.

			For n=1, there are no arrangements with two black tiles touching as there is only space for one tile.
n=2 is a 2 X 2 grid, and there are 9 possible arrangements where at least two black tiles touch on an edge:
  00  01  10  11  01  10  11  11  11
  11, 01, 10, 00, 11, 11, 01, 10, 11.
n=3 is a 3 X 3 grid, and with 2^9=512 possible combinations of black and white tiles, 449 of them have at least two black tiles touching.

Cf. A002416, A006506.

a(n) = 2^(n^2) - A006506(n).

A383255 Number of n X n {0,1,2,3} matrices having no 1's to the right of any 0's and no 3's above any 2's.

Original entry on oeis.org

1, 4, 194, 107080, 672498596, 48104236145168, 39202958861329453384, 364022757339778569993689888, 38513979937284562006371342202842000, 46429021191757554279412904483559912259714112, 637737721080296383894709847744103523361428384973270816

Offset: 0

John Tyler Rascoe, Apr 20 2025

nonn

These are matrices with no [0,1] or [3] submatrices.

[2]

			The 3 X 3 matrices below are counted under a(3) = 107080:
 [0,0,0] [1,0,2] [2,3,2]
 [0,0,0] [1,0,3] [3,3,3]
 [0,0,0],[0,2,3],[3,3,3].

John Tyler Rascoe, Python program.

Cf. A002416, A006506, A014235, A060757, A181213, A213977, A381857.

Python
```
# see links
```

a(n) <= A060757(n).

a(5)-a(10) from Bert Dobbelaere, Apr 23 2025

A068472 Number of n X n binary matrices with every 1 having at most 3 adjacent 1's.

Original entry on oeis.org

1, 2, 16, 496, 58640, 26536192, 45851039232, 302758305892480, 7638804476736307712, 736437724731312162567680, 271287639195997221896855543808, 381862430868672544566361613406502912

Offset: 0

R. H. Hardin, Mar 10 2002

nonn

No adjacent 1's A006506, one adjacent 1 A067968.

a(0)=1 prepended by Alois P. Heinz, Sep 26 2024

cp's OEIS Frontend

A068471 Number of n X n binary matrices with each 1 having at most 2 adjacent 1's.

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A201222 Number of ways to place k non-attacking knights on a 2 X n horizontal cylinder, summed over all k>=0.

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Mathematica

A283161 Natural numbers whose digits can be formed by typing non-adjacent keys on a 123-456-789 keypad without repeating a digit.

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Mathematica

A350336 Number of n X n ternary matrices with no two adjacent 0's.

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Mathematica

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A381857 Number of n X n binary matrices with at least 2 adjacent 1's.

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A383255 Number of n X n {0,1,2,3} matrices having no 1's to the right of any 0's and no 3's above any 2's.

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Comments

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Python

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Extensions

A068472 Number of n X n binary matrices with every 1 having at most 3 adjacent 1's.

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Author

Keywords

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Extensions